A previous discussion on a more intuitive variant of a UFD led naturally to discussing signs on commutative monoids, where instead of thinking of signs as additive inverses in a ring, we generalized to defining a more abstract concept of sign. Upon further thought, we can probably generalize this definition even further, from a commutative monoid to a most general kind of algebra — namely, a magma. In this post, we take this approach, which has a “reverse mathematics” benefit of seeing the most general settings in which certain theorems hold and certain questions can be formulated.
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be an infinite set. Then, there exists a partition of 
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